Finite Group Databasebeta

← #30648183

Subgroups — #30648183

Order 16256.

496 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16861231C211yesmaximal, characteristic, complemented
2C2×#16861229C211yesmaximal, characteristic, complemented
3#22550422C211yesmaximal, characteristic, complemented
4#22550344C211yesmaximal, characteristic, complemented
5#22550101C211yesmaximal, characteristic, complemented
6#22550097C211yesmaximal, characteristic, complemented
7#22550094C211yesmaximal, characteristic, complemented
8#22550090C211yesmaximal, characteristic, complemented
9#22550085C211yesmaximal, characteristic, complemented
10#22550080C211yesmaximal, characteristic, complemented
11#22550075C211yesmaximal, characteristic, complemented
12#22550070C211yesmaximal, characteristic, complemented
13C2×Dic127⋊C8C211yesmaximal, characteristic, complemented
14#22549870C211yesmaximal, characteristic, complemented
15#416×C127C211yesmaximal, characteristic, complemented
16C2×#14523248C2211yescharacteristic
17C2×#14523248C2211yescharacteristic
18#16861231—24no—
19#16861229—24no—
20C2×Dic508C2211yescharacteristic
21C2×#14523238C2211yescharacteristic
22C2×#14523237C2211yescharacteristic
23C2×#14523237C2211yescharacteristic
24C2×#14523236C2211yescharacteristic
25C2×#14523235C2211yescharacteristic
26C2×D1016C2211yescharacteristic
27C2×#14523233C2211yescharacteristic
28C22×C127⋊C8C2211yescharacteristic
29#16861207C2211yescharacteristic
30#16861205C2211yescharacteristic
31#16861198C2211yescharacteristic
32#16861196C2211yescharacteristic
33#16861181—21no—
34#16861181—21no—
35#16861174C2211yescharacteristic
36#16861174C2211yescharacteristic
37#16861171C2211yescharacteristic
38#16861171C2211yescharacteristic
39C4×C22⋊D127—21no—
40C4×C22⋊D127—21no—
41(C22⋊D127)⋊C4—21no—
42(C22⋊D127)⋊C4—21no—
43#16861166—21no—
44C2×C4×Dic127C2211yescharacteristic
45C42×D127—21no—
46Dic127⋊C8C2211yescharacteristic
47Dic127⋊C8C2211yescharacteristic
48Dic127⋊C8C2211yescharacteristic
49Dic127⋊C8C2211yescharacteristic
50#16861132C2211yescharacteristic
51#16861132C2211yescharacteristic
52#16861129C2211yescharacteristic
53#16861129C2211yescharacteristic
54C2×#41×C127C2211yescharacteristic
55C22×C8×C127C2211yescharacteristic
56#117×C127C2211yescharacteristic
57#103×C127C2211yescharacteristic
58#103×C127C2211yescharacteristic
59#102×C127C2211yescharacteristic
60#102×C127C2211yescharacteristic
61#14523248—22no—
62#14523248—22no—
63#14523248—22no—
64#14523248D414yescomplemented
65C2×Dic254C2311yescharacteristic
66C2×Dic254C2311yescharacteristic
67C2×D508C2311yescharacteristic
68C2×D508C2311yescharacteristic
69C2×C22⋊D127—21no—
70C2×C22⋊D127—21no—
71C2×C22⋊D127—21no—
72C2×C22⋊D127—21no—
73C22×Dic127D411yescharacteristic
74C22×Dic127D411yescharacteristic
75C2×C4×D127—21no—
76C2×C4×D127—21no—
77C2×C4×D127—21no—
78C2×C4×D127—21no—
79Dic508—22no—
80#14523238—22no—
81#14523237—22no—
82#14523237—22no—
83#14523236—22no—
84#14523235—22no—
85D1016—22no—
86#14523233—22no—
87C2×C127⋊C8—22no—
88C2×C127⋊C8C2311yescharacteristic
89C2×C127⋊C8C2311yescharacteristic
90Dic127⋊C4—21no—
91Dic127⋊C4—21no—
92Dic127⋊C4—21no—
93Dic127⋊C4—21no—
94C22⋊Dic127—21no—
95C22⋊Dic127—21no—
96(C2×D127)⋊C4—21no—
97(C2×D127)⋊C4—21no—
98(C2×D127)⋊C4—21no—
99(C2×D127)⋊C4—21no—
100C4×Dic127—21no—
101C4×Dic127—21no—
102C4×Dic127D411yescharacteristic
103C4×Dic127D411yescharacteristic
104C4×Dic127C2311yescharacteristic
105C4×Dic127C2311yescharacteristic
106#41×C127—22no—
107#41×C127—22no—
108C2×Q8×C127C2311yescharacteristic
109C2×D4×C127—21no—
110C2×D4×C127C2311yescharacteristic
111C22×C4×C127—21no—
112C22×C4×C127C2311yescharacteristic
113C2×C8×C127—22no—
114C2×C8×C127C2311yescharacteristic
115C2×C8×C127C2311yescharacteristic
116C4⋊C4×C127C2311yescharacteristic
117C4⋊C4×C127C2311yescharacteristic
118C22⋊C4×C127—21no—
119C42×C127—21no—
120C22×D127—21no—
121C22×D127—21no—
122Dic254—21no—
123Dic254—21no—
124Dic254—21no—
125Dic254C2×D412yes—
126D508—21no—
127D508—21no—
128D508—21no—
129D508C2×D412yes—
130C22⋊D127—24no—
131C22⋊D127—42no—
132C22⋊D127—42no—
133C22⋊D127—42no—
134C2×Dic127—21no—
135C2×Dic127—22no—
136C2×Dic127—21no—
137C2×Dic127—41no—
138C2×Dic127C2×D411yescharacteristic
139C2×Dic127C2×D411yescharacteristic
140C2×Dic127C2×D411yescharacteristic
141C2×Dic127C2×D411yescharacteristic
142C4×D127—24no—
143C4×D127—42no—
144C4×D127—42no—
145C4×D127—42no—
146C127⋊C8—21no—
147C127⋊C8—21no—
148C23×C127—21no—
149C23×C127C2×D411yescharacteristic
150Q8×C127—21no—
151Q8×C127—21no—
152D4×C127—21no—
153D4×C127—21no—
154D4×C127—42no—
155C2×C4×C127—21no—
156C2×C4×C127—21no—
157C2×C4×C127—21no—
158C2×C4×C127—42no—
159C2×C4×C127C2×D412yes—
160C2×C4×C127C2×D411yescharacteristic
161C2×C4×C127#4112yes—
162C2×C4×C127C2411yescharacteristic
163C8×C127—21no—
164C8×C127—21no—
165C22×C127—21no—
166C22×C127—21no—
167C22×C127—41no—
168C22×C127—41no—
169C22×C127—41no—
170C22×C127#12112yes—
171C22×C127C22×D411yescharacteristic
172C2×D127—22no—
173C2×D127—41no—
174C2×D127—41no—
175C2×D127—41no—
176C2×D127—41no—
177C2×D127—41no—
178C4×C127—41no—
179C4×C127—41no—
180C4×C127—41no—
181C4×C127#12112yes—
182C4×C127C22×D411yescharacteristic
183C4×C127C2×#4111yescharacteristic
184Dic127—22no—
185Dic127—21no—
186Dic127—21no—
187Dic127—21no—
188Dic127—41no—
189Dic127#13512yescomplemented
190C2×C127—22no—
191C2×C127—81no—
192C2×C127C2×#12111yescharacteristic
193C2×C127C2×#13511yescharacteristic
194C2×C127#57511yescharacteristic
195D127—42no—
196D127—81no—
197#3035—1271nomaximal, Sylow
198C127#303511yescharacteristic, complemented, Sylow
199C2×#135—1271no—
200C2×#135—1271no—
201#532—1271no—
202C4×#41—1271no—
203#464—1271no—
204#464—1271no—
205#463—1271no—
206#463—1271no—
207#462—1271no—
208#461—1271no—
209#460—1271no—
210#459—1271no—
211C2×C4⋊C8—1271no—
212#416—1271no—
213#416D12711yescharacteristic, complemented
214C2×#41—1271no—
215C2×#41—1271no—
216C2×#41C2×D12711yescharacteristic
217#135—2544no—
218#135—2544no—
219C2×Q16—1271no—
220C2×Q16—1271no—
221C2×SD16—1271no—
222C2×SD16—1271no—
223C2×SD16—1271no—
224C2×SD16—1271no—
225C2×D8—1271no—
226C2×D8—1271no—
227C22×C8—1271no—
228C22×C8C2×D12711yescharacteristic
229C4⋊Q8—1271no—
230#127—1271no—
231#124—1271no—
232#124—1271no—
233#121—2541no—
234#121—2541no—
235C4×Q8—1271no—
236C4×Q8—1271no—
237C4×D4—1271no—
238C4×D4—1271no—
239C4×D4—2541no—
240C4×D4—2541no—
241C4×D4—2541no—
242C4×D4—2541no—
243#117—2541no—
244#117C2×D12711yescharacteristic
245C2×C42—1271no—
246C2×C42—2541no—
247C4⋊C8—1271no—
248C4⋊C8—1271no—
249C4⋊C8—1271no—
250C4⋊C8—1271no—
251#103—1271no—
252#103—1271no—
253#103C2×D12711yescharacteristic
254#103C2×D12711yescharacteristic
255#102—1271no—
256#102—1271no—
257#102C2×D12711yescharacteristic
258#102C2×D12711yescharacteristic
259#41—22no—
260#41—22no—
261#41—1274no—
262#41—2542no—
263#41—2542no—
264#41—2542no—
265C2×Q8—1271no—
266C2×Q8—1271no—
267C2×Q8C22×D12711yescharacteristic
268C2×D4—21no—
269C2×D4—1271no—
270C2×D4—1271no—
271C2×D4—2541no—
272C2×D4—2541no—
273C2×D4—2541no—
274C2×D4—2541no—
275C2×D4C22×D12711yescharacteristic
276C22×C4—21no—
277C22×C4—1271no—
278C22×C4—1271no—
279C22×C4—2541no—
280C22×C4—2541no—
281C22×C4—2541no—
282C22×C4—2541no—
283C22×C4C22×D12711yescharacteristic
284Q16—2542no—
285Q16—2542no—
286SD16—2542no—
287SD16—2542no—
288SD16—2542no—
289SD16—2542no—
290D8—2542no—
291D8—2542no—
292C2×C8—22no—
293C2×C8—1271no—
294C2×C8—1271no—
295C2×C8—2542no—
296C2×C8C22×D12711yescharacteristic
297C2×C8C22×D12711yescharacteristic
298C4⋊C4—2541no—
299C4⋊C4—2541no—
300C4⋊C4—2541no—
301C4⋊C4—2541no—
302C4⋊C4C22×D12711yescharacteristic
303C4⋊C4C22×D12711yescharacteristic
304C22⋊C4—21no—
305C22⋊C4—2541no—
306C22⋊C4—2541no—
307C22⋊C4—2541no—
308C22⋊C4—2541no—
309C22⋊C4—2541no—
310C22⋊C4—2541no—
311C42—21no—
312C42—1271no—
313C42—1271no—
314C42—1271no—
315C42—1271no—
316C42—2541no—
317C42—2541no—
318C23—21no—
319C23—2541no—
320C23—2541no—
321C23D4×D12711yescharacteristic
322Q8—21no—
323Q8—21no—
324Q8—1272no—
325Q8—2541no—
326Q8—2541no—
327Q8—2541no—
328D4—21no—
329D4—21no—
330D4—42no—
331D4—1272no—
332D4—5082no—
333D4—5082no—
334D4—5082no—
335D4—2544no—
336D4—2541no—
337D4—2541no—
338D4—2541no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—42no—
343C2×C4—1271no—
344C2×C4—1271no—
345C2×C4—1271no—
346C2×C4—1271no—
347C2×C4—5082no—
348C2×C4—5082no—
349C2×C4—5082no—
350C2×C4—5081no—
351C2×C4—2542no—
352C2×C4—2541no—
353C2×C4—2541no—
354C2×C4—2544no—
355C2×C4D4×D12712yes—
356C2×C4D4×D12711yescharacteristic
357C2×C4#1452324812yes—
358C2×C4C23×D12711yescharacteristic
359C8—21no—
360C8—21no—
361C8—2541no—
362C8—2541no—
363C22—21no—
364C22—21no—
365C22—41no—
366C22—41no—
367C22—41no—
368C22—5081no—
369C22—5081no—
370C22—5081no—
371C22—5081no—
372C22—5081no—
373C22—2542no—
374C22#1686118212yes—
375C22C2×D4×D12711yescharacteristic
376C4—41no—
377C4—41no—
378C4—41no—
379C4—1272no—
380C4—5081no—
381C4—2542no—
382C4—2541no—
383C4—2541no—
384C4—2541no—
385C4#1686118212yes—
386C4C2×D4×D12711yescharacteristic
387C4C2×#1452324811yescharacteristic
388C2—22no—
389C2—81no—
390C2—10161no—
391C2—5082no—
392C2C2×#1686118211yescharacteristic
393C2#135×D12711yescharacteristic
394C2#2255067311yescharacteristic
subgroupG/H
centerC2×C4D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#416×C127C2
socleC22×C127C22×D4
hypercenter#416D127
hypocenterC127#3035
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ #416

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×#16861231C2semidirect
2. C2×#16861229C2semidirect
3. #22550422C2semidirect
4. #22550344C2semidirect
5. #22550101C2semidirect
6. #22550097C2semidirect
7. #22550094C2semidirect
8. #22550090C2semidirect
9. #22550085C2semidirect
10. #22550080C2semidirect
11. #22550075C2semidirect
12. #22550070C2semidirect
13. C2×Dic127⋊C8C2semidirect
14. #22549870C2semidirect
15. #416×C127C2semidirect
64. #14523248D4semidirect
189. Dic127#135semidirect
198. C127#3035semidirect
213. #416D127semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#168612312
C2×#168612292
#225504222
#225503442
#225501012
#225500972
#225500942
#225500902
#225500852
#225500802
#225500752
#225500702
C2×Dic127⋊C82
#225498702
#416×C1272
#3035127